Question: In the figure shown, arc $ADB$ and arc $BEC$ are semicircles, each with a radius of one unit. Point $D$, point $E$ and point $F$ are the midpoints of arc $ADB$, arc $BEC$ and arc $DFE$, respectively. If arc $DFE$ is also a semicircle, what is the area of the shaded region?

[asy]
unitsize(0.5inch);
path t=(1,1)..(2,0)--(0,0)..cycle;
draw(t);
path r=shift((2,0))*t;
path s=shift((1,1))*t;
draw(s);
fill(s,gray(0.7));
fill((1,0)--(1,1)--(3,1)--(3,0)--cycle,gray(0.7));
fill(t,white);
fill(r,white);
draw(t);
draw(r);
dot((0,0));
dot((1,1));
dot((2,2));
dot((3,1));
dot((2,0));
dot((4,0));
label("$A$",(0,0),W);
label("$B$",(2,0),S);
label("$C$",(4,0),E);
label("$D$",(1,1),NW);
label("$E$",(3,1),NE);
label("$F$",(2,2),N);
[/asy]
Answer: Mark segments $BF$ and $DE$ and label their intersection $P$.  Now, slide sector $FPE$ on to sector $DB$ and slide sector $FPD$ on to sector $EB$.  The shaded region is now a rectangle with side lengths $1$ and $2$, thus the area of the shaded region is $\boxed{2}$.